Beckham's current contract with Los Angeles Galaxy expires later this year and he has yet to decide whether to continue his stay with the American outfit or accept one of several offers to return to Europe.
The 36-year-old former Manchester United and Real Madrid star is keen to play for Great Britain at the London Olympics next year and that could tempt him to return to England.
Tottenham and QPR are believed to be interested in Beckham, but Eriksson has always been close to the midfielder, who captained England during his tenure as manager, and he is trying to persuade him to join the ambitious English second tier club.
"I have already had that discussion with him. I spoke to him about Leicester," Eriksson told The Times.
"Beckham, he will never say no to me - he says, 'Sven, we'll see, we'll see'.
"The big thing is what he wants to do when his contract finishes in LA. Will he want to go on playing?"
Eriksson knows Leicester, who have substantial investment from their Thai owners, will face stiff competition from Paris St Germain, who are also reported to be interested.
But the Swede hopes the added incentive of a training role will help tip the balance.
"We will give him a whistle and he can go out (on the training pitch) straight away," he said.
Meanwhile Eriksson, who was the first overseas manager to take charge of England, believes Arsenal boss Arsene Wenger could be tempted to succeed Fabio Capello when the Italian steps down after Euro 2012.
Wenger has suffered a troubled time at Arsenal over the last year and Eriksson wouldn't be surprised if he was tempted to take the England job.
"Reporters talk about (Harry) Redknapp as the people's choice and I'm quite sure if they could get Wenger, they would take him," Eriksson said.
"Arsene Wenger can adapt to everything. He lives for football.
"You see how much it matters to him every game, but you also see when it's a European Championship or a World Cup, he is working on it, for French television, so he doesn't miss anything. That's his holiday."
Source: AFP
Source: AFP